FLORENTIN SMARANDACHE Counter-Projective Geometry

In Florentin Smarandache: “Collected Papers”, vol. II. Chisinau (Moldova): Universitatea de Stat din Moldova, 1997.

the assumption of the fifth postulate is ful:filled, but the consequence does not hold,

because h, and hz do not cut each other (they may not be extended beyond A and B

respectively, because the lines would not be geodesics anymore).

Question 29

Find a more convincing midel for tbis non-geometry.

COUNTER-PROJECTIVE GEOMETRY

Let P, L be two sets, and r a relation included in P x 1. The elements of P are called

points, and those of L lines. When (p, I) belongs to r, we say that the line l contains the point

p. For these, one imposes the following COl:NTER-AXIOMS:

(I) There exist: either at least two lines, or no line, that contains two given distinct

points.

(II) Let p"P"P3 be three non-collinear points, and qJ,qz two distinct points. Supoose

that {PI, q,;P3} and {P2, q"P3} are collinear triples. Then the line containing P}'Pz,

and the Dne containing ql, q2 do not intersect.

(III) Every line contains at most two distinct points.

Questions 30-31:

Find a model for the Counter-(General Projective) Geometry (the previous I and II counter­

axioms hold), and a model for the Counter-Projective Geometry (the previous I, II, and III

counter-axioms hold). [They are called COl:NTER-MODELS for the general projective, and

projective geometry, respectively.]

Questions 32-33:

Find geometric modis for each of the following two cases:

- There are points/lines that verify all the previous counter-axioms, and other

points/lines in the same COUNTER-PROJECTIVE SPACE that do not verify any

of them;

- Some of the counter-axioms I, II, III are verified, while the others are not (there are

particular cases already known).

Question 34:

The study of these counter-models may be extended to Infinite-Dimensional Real (or Com­

plex) Projective Spaces, denying the IV-th axioms, i.e.:

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(IV) There exists no set of finite number of points for which any subspace that

contains all of them contains P.

Question 35:

Does the Duality Principle hold in a counter-projective space?

What about Desargues's Theorem, Fundamental Theorem of Projective Geometry /Theorem

of Pappus, and Staudt Algebra?

Or Pascal's Theorem, Brianchon's Theorem? (1 think none of them will hold!)

Question 36:

The theory of Buildings of Tits, which contains the Projective Geometry as a particular

case, can be 'distorted' in the same <paracioxist> way by deforming its axiom of a BN-pair (or

Tits system) for the triple (G, B, N), where G is a group, and B, ri its subgroups; [see J.Tits,

"Buildings of spnerical type and finite B .... -pairs", Lecture notes in math. 386, Springer, 1974}.

riot ions as: simplex, complex, chamber, codimension, apartment, building will get contorted

either ...

Develop a Theory of Distorted Buldings of Tits!

ANTI-GEOMETRY

It is possible to de-formalize entirely Hilbert's groups of axioms of the Euclidean Geometry,

and to construct a model such that none of his fixed axiom holds.

Let's consider the following things:

and

- a set of <points>: A, B, C, ...

- a set of <lines>: h, k, I, .. .

-a set of <planes>; 0.,13", .. .

- a set of relationship among these elements: "are situated", "between", "parallel",

"congruent", "continous", etc.

Then, we can deny all Hilbert's twenty axioms [see David Hilbert, "Foundation of Ge­

ometry", translated by E.J.Towsend, 1950; and Roberto Bonola, "Non-Euclidean Geometry",

1938]. There exist casses, whithin a geometric model, when the same axiom is verifyed by

certain points/lines/planes and denied by others.

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